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Matrix Multiplication Verification Using Coding Theory

Authors Huck Bennett , Karthik Gajulapalli , Alexander Golovnev , Evelyn Warton

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Author Details

Huck Bennett
  • University of Colorado Boulder, CO, USA
Karthik Gajulapalli
  • Georgetown University, Washington DC, USA
Alexander Golovnev
  • Georgetown University, Washington DC, USA
Evelyn Warton
  • Oregon State University, Corvallis, OR, USA

Funding

  • Bennett, Huck: Supported by NSF Grant CCF-2312297.
  • Gajulapalli, Karthik: Supported by NSF grant CCF-2338730.
  • Golovnev, Alexander: Supported by NSF grant CCF-2338730.

Acknowledgements

We thank Amir Nayyeri for many helpful discussions in the early stages of work on this paper, and Mark Iwen [Mark A. Iwen, 2023] for answering questions about [Mark A. Iwen and Craig V. Spencer, 2009]. We also thank the anonymous reviewers for their helpful comments.

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Huck Bennett, Karthik Gajulapalli, Alexander Golovnev, and Evelyn Warton. Matrix Multiplication Verification Using Coding Theory. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.42

Abstract

We study the Matrix Multiplication Verification Problem (MMV) where the goal is, given three n × n matrices A, B, and C as input, to decide whether AB = C. A classic randomized algorithm by Freivalds (MFCS, 1979) solves MMV in Õ(n²) time, and a longstanding challenge is to (partially) derandomize it while still running in faster than matrix multiplication time (i.e., in o(n^ω) time).
To that end, we give two algorithms for MMV in the case where AB - C is sparse. Specifically, when AB - C has at most O(n^δ) non-zero entries for a constant 0 ≤ δ < 2, we give (1) a deterministic O(n^(ω-ε))-time algorithm for constant ε = ε(δ) > 0, and (2) a randomized Õ(n²)-time algorithm using δ/2 ⋅ log₂ n + O(1) random bits. The former algorithm is faster than the deterministic algorithm of Künnemann (ESA, 2018) when δ ≥ 1.056, and the latter algorithm uses fewer random bits than the algorithm of Kimbrel and Sinha (IPL, 1993), which runs in the same time and uses log₂ n + O(1) random bits (in turn fewer than Freivalds’s algorithm).
Our algorithms are simple and use techniques from coding theory. Let H be a parity-check matrix of a Maximum Distance Separable (MDS) code, and let G = (I | G') be a generator matrix of a (possibly different) MDS code in systematic form. Our deterministic algorithm uses fast rectangular matrix multiplication to check whether HAB = HC and H(AB)^T = H(C^T), and our randomized algorithm samples a uniformly random row g' from G' and checks whether g'AB = g'C and g'(AB)^T = g'C^T.
We additionally study the complexity of MMV. We first show that all algorithms in a natural class of deterministic linear algebraic algorithms for MMV (including ours) require Ω(n^ω) time. We also show a barrier to proving a super-quadratic running time lower bound for matrix multiplication (and hence MMV) under the Strong Exponential Time Hypothesis (SETH). Finally, we study relationships between natural variants and special cases of MMV (with respect to deterministic Õ(n²)-time reductions).

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Error-correcting codes
Keywords
  • Matrix Multiplication Verification
  • Derandomization
  • Sparse Matrices
  • Error-Correcting Codes
  • Hardness Barriers
  • Reductions

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